näyteautokorrelaatio

Sample autocorrelation, known in Finnish as näyteautokorrelaatio, is a fundamental concept in time series analysis. It is a statistical measure that quantifies the degree of similarity between a time series and a lagged version of itself. In simpler terms, it tells us how correlated observations in a time series are with observations at earlier points in time.

The sample autocorrelation function (SACF) is typically represented as a plot, with the x-axis indicating the lag and the y-axis showing the autocorrelation coefficient. A positive autocorrelation at a given lag suggests that high values in the series tend to be followed by high values at that lag, and low values by low values. Conversely, a negative autocorrelation implies that high values are followed by low values, and vice versa. A value close to zero indicates little to no linear relationship.

The sample autocorrelation is calculated for discrete time series data. For a time series $X = \{x_1, x_2, ..., x_n\}$, the sample autocorrelation at lag $k$ is typically estimated as:

$r_k = \frac{\sum_{t=1}^{n-k}(x_t - \bar{x})(x_{t+k} - \bar{x})}{\sum_{t=1}^{n}(x_t - \bar{x})^2}$

where $\bar{x}$ is the sample mean of the time series.

Understanding the sample autocorrelation is crucial for identifying the underlying structure of a time series. It helps in selecting appropriate models for forecasting and analysis, such as autoregressive (AR), moving average (MA), or ARIMA models. For instance, a sharp drop in autocorrelation after a few lags might suggest an MA process, while a slow decay could indicate an AR process.